On the automorphism group of a groupoid
For a groupoid \((X,\cdot)\) (in the sense of a set with a binary operation), the authors study its automorphism group as a permutation group of~\(X\) endowed with a family \((\sigma_i)_{i\in I}\) of transformations of \(X\) associated with the orbits \((X_i)_{i\in I}\) such that each \(\sigma_i\) commutes with the action of the stabilizer of \(X_i\). Conversely, they show how to associate with a faithful action of a group \(G\) on a set \(X\), together with a family of transformations of~\(X\) as above, a groupoid structure on~\(X\) in whose automorphism group \(G\) embeds naturally. Moreover, if the group was constructed as above from a groupoid, then the groupoid thus obtained is the original one. The paper goes on to analyze how various special properties on the groupoid relate to special properties of the group actions. In particular, they obtain a proof of a theorem of \textit{M. Gould} [Algebra Univers. 2, 54-56 (1972; Zbl 0248.20071)] stating that an at most countable group \(G\) has a left-cancellable groupoid structure whose automorphism group is isomorphic to~\(G\).
scientific article; zbMATH DE number 1275893
scientific article